# Calculus/Limits/Exercises

< Calculus‎ | Limits
 ← Proofs of Some Basic Limit Rules Calculus Differentiation → Limits/Exercises

## Basic Limit Exercises

1. $\lim _{x\to 2}{\Big [}4x^{2}-3x-1{\Big ]}$
$9$
$9$
2. $\lim _{x\to 5}{\Big [}x^{2}{\Big ]}$
$25$
$25$

## One-Sided Limits

Evaluate the following limits or state that the limit does not exist.

3. $\lim _{x\to 0^{-}}{\frac {x^{3}+x^{2}}{x^{3}+2x^{2}}}$
${\frac {1}{2}}$
${\frac {1}{2}}$
4. $\lim _{x\to 7^{-}}{\Big [}|x^{2}+x|-x{\Big ]}$
$49$
$49$
5. $\lim _{x\to -1^{+}}{\sqrt {1-x^{2}}}$
$0$
$0$
6. $\lim _{x\to -1^{-}}{\sqrt {1-x^{2}}}$
The limit does not exist
The limit does not exist

## Two-Sided Limits

Evaluate the following limits or state that the limit does not exist.

7. $\lim _{x\to -1}{\frac {1}{x-1}}$
$-{\frac {1}{2}}$
$-{\frac {1}{2}}$
8. $\lim _{x\to 4}{\frac {1}{x-4}}$
The limit does not exist.
The limit does not exist.
9. $\lim _{x\to 2}{\frac {1}{x-2}}$
The limit does not exist.
The limit does not exist.
10. $\lim _{x\to -3}{\frac {x^{2}-9}{x+3}}$
$-6$
$-6$
11. $\lim _{x\to 3}{\frac {x^{2}-9}{x-3}}$
$6$
$6$
12. $\lim _{x\to -1}{\frac {x^{2}+2x+1}{x+1}}$
$0$
$0$
13. $\lim _{x\to -1}{\frac {x^{3}+1}{x+1}}$
$3$
$3$
14. $\lim _{x\to 4}{\frac {x^{2}+5x-36}{x^{2}-16}}$
${\frac {13}{8}}$
${\frac {13}{8}}$
15. $\lim _{x\to 25}{\frac {x-25}{{\sqrt {x}}-5}}$
$10$
$10$
16. $\lim _{x\to 0}{\frac {|x|}{x}}$
The limit does not exist.
The limit does not exist.
17. $\lim _{x\to 2}{\frac {1}{(x-2)^{2}}}$
$\infty$
$\infty$
18. $\lim _{x\to 3}{\frac {\sqrt {x^{2}+16}}{x-3}}$
The limit does not exist.
The limit does not exist.
19. $\lim _{x\to -2}{\frac {3x^{2}-8x-3}{2x^{2}-18}}$
$-{\frac {5}{2}}$
$-{\frac {5}{2}}$
20. $\lim _{x\to 2}{\frac {x^{2}+2x+1}{x^{2}-2x+1}}$
$9$
$9$
21. $\lim _{x\to 3}{\frac {x+3}{x^{2}-9}}$
The limit does not exist.
The limit does not exist.
22. $\lim _{x\to -1}{\frac {x+1}{x^{2}+x}}$
$-1$
$-1$
23. $\lim _{x\to 1}{\frac {1}{x^{2}+1}}$
${\frac {1}{2}}$
${\frac {1}{2}}$
24. $\lim _{x\to 1}\left[x^{2}+5x-{\frac {1}{2-x}}\right]$
$5$
$5$
25. $\lim _{x\to 1}{\frac {x^{2}-1}{x^{2}+2x-3}}$
${\frac {1}{2}}$
${\frac {1}{2}}$
26. $\lim _{x\to 1}{\frac {5x}{x^{2}+2x-3}}$
The limit does not exist.
The limit does not exist.

## Limits to Infinity

Evaluate the following limits or state that the limit does not exist.

27. $\lim _{x\to \infty }{\frac {-x+\pi }{x^{2}+3x+2}}$
$0$
$0$
28. $\lim _{x\to -\infty }{\frac {x^{2}+2x+1}{3x^{2}+1}}$
${\frac {1}{3}}$
${\frac {1}{3}}$
29. $\lim _{x\to -\infty }{\frac {3x^{2}+x}{2x^{2}-15}}$
${\frac {3}{2}}$
${\frac {3}{2}}$
30. $\lim _{x\to -\infty }{\Big [}3x^{2}-2x+1{\Big ]}$
$\infty$
$\infty$
31. $\lim _{x\to \infty }{\frac {2x^{2}-32}{x^{3}-64}}$
$0$
$0$
32. $\lim _{x\to \infty }6$
$6$
$6$
33. $\lim _{x\to \infty }{\frac {3x^{2}+4x}{x^{4}+2}}$
$0$
$0$
34. $\lim _{x\to -\infty }{\frac {2x+3x^{2}+1}{2x^{2}+3}}$
${\frac {3}{2}}$
${\frac {3}{2}}$
35. $\lim _{x\to -\infty }{\frac {x^{3}-3x^{2}+1}{3x^{2}+x+5}}$
$-\infty$
$-\infty$
36. $\lim _{x\to \infty }{\frac {x^{2}+2}{x^{3}-2}}$
$0$
$0$

## Limits of Piecewise Functions

Evaluate the following limits or state that the limit does not exist.

37. Consider the function

$f(x)={\begin{cases}(x-2)^{2}&{\text{if }}x<2\\x-3&{\text{if }}x\geq 2\end{cases}}$
a. $\lim _{x\to 2^{-}}f(x)$
$0$
$0$
b. $\lim _{x\to 2^{+}}f(x)$
$-1$
$-1$
c. $\lim _{x\to 2}f(x)$
The limit does not exist
The limit does not exist

38. Consider the function

$g(x)={\begin{cases}-2x+1&{\text{if }}x\leq 0\\x+1&{\text{if }}0
a. $\lim _{x\to 4^{+}}g(x)$
$18$
$18$
b. $\lim _{x\to 4^{-}}g(x)$
$5$
$5$
c. $\lim _{x\to 0^{+}}g(x)$
$1$
$1$
d. $\lim _{x\to 0^{-}}g(x)$
$1$
$1$
e. $\lim _{x\to 0}g(x)$
$1$
$1$
f. $\lim _{x\to 1}g(x)$
$2$
$2$

39. Consider the function

$h(x)={\begin{cases}2x-3&{\text{if }}x<2\\8&{\text{if }}x=2\\-x+3&{\text{if }}x>2\end{cases}}$
a. $\lim _{x\to 0}h(x)$
$-3$
$-3$
b. $\lim _{x\to 2^{-}}h(x)$
$1$
$1$
c. $\lim _{x\to 2^{+}}h(x)$
$1$
$1$
d. $\lim _{x\to 2}h(x)$
$1$
$1$